Universal arrows from an object to a functor and from a functor to an object

Let $U:\mathcal{D}\to \mathcal{C}$ be a functor as in Covariant functor, identity functor, composite functor, and contravariant functor and let $X$ be an object of $\mathcal{C}$.

A universal arrow from $X$ to $U$ is a pair $(R,\eta )$ with $R\in \mathcal{D}$ and $\eta :X\to U(R)$ such that, for every $D\in \mathcal{D}$ and $f:X\to U(D)$, there is a unique $h:R\to D$ satisfying

$f=U(h)\circ \eta \mathrm{.}$

When $\mathcal{C}$ is locally small, the relevant hom-assignment is a functor by The assignments $\mathcal{C}(a,-)$ and $\mathcal{C}(-,a)$ are functors to $\mathbf{S}\mathbf{e}\mathbf{t}$, and equivalently $(R,\eta )$ is a universal element of the covariant functor $\mathcal{C}(X,U(-)):\mathcal{D}\to \mathbf{S}\mathbf{e}\mathbf{t}$ in the sense of Universal elements of covariant functors and presheaves.

A universal arrow from $U$ to $X$ is a pair $(R,\epsilon )$ with $\epsilon :U(R)\to X$ such that, for every $D\in \mathcal{D}$ and $f:U(D)\to X$, there is a unique $h:D\to R$ satisfying

$f=\epsilon \circ U(h)\mathrm{.}$

When $\mathcal{C}$ is locally small, equivalently $(R,\epsilon )$ is a universal element of the presheaf $\mathcal{C}(U(-),X):{\mathcal{D}}^{\mathrm{o}\mathrm{p}}\to \mathbf{S}\mathbf{e}\mathbf{t}$. The associated comma categories $(X\downarrow U)$ and $(U\downarrow X)$ are those of Comma category, slice category, and coslice category.